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A000581
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Binomial coefficients C(n,8).
(Formerly M4626 N1976)
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43
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1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, 7888725, 10518300, 13884156, 18156204, 23535820, 30260340, 38608020, 48903492, 61523748, 76904685
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OFFSET
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8,2
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COMMENTS
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Figurate numbers based on 8-dimensional regular simplex. - Jonathan Vos Post, Nov 28 2004
a(n) = A110555(n+1,8). - Reinhard Zumkeller, Jul 27 2005
Just as A005712 and A000574 are described as the coefficients of x^4 and x^5 in the expansion of (1+x+x^2)^n, so should this sequence be described as the coefficients of x^3 therein. - R. K. Guy, Oct 19 2007
Product of 8 consecutive numbers divided by 8!. - Artur Jasinski, Dec 02 2007
In this sequence there are no primes. - Artur Jasinski, Dec 02 2007
a(n) = number of (n-8)-digit numbers with nondescending digits. E.g., a(9) = 9 = {1,2,3,..,9}, a(10) = 45 = {11-19, 22-29, 33-39, ..., 99} [0 is counted as a zero-digit number rather than a 1-digit number]. - Toby Gottfried, Feb 14 2012
Sum_{k>=8} 1/a(k) = 8/7. - Tom Edgar, Sep 10 2015
a(n) =fallfac(n, 8)/8! = binomial(n, 8) is also the number of independent components of an antisymmetric tensor of rank 8 and dimension n >= 8 (for n = 1..7 this becomes 0). Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 9 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-8 into exactly 9 parts. - Juergen Will, Jan 02 2016
Partial sums of A000580. - Art Baker, Mar 26 2019
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 8..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 258
H. K. Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc. 131 (2003), 65-75.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - Juergen Will, Jan 02 2016
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Composition
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FORMULA
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G.f.: x^8/(1-x)^9.
a(n) = n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)/8!. - Artur Jasinski, Dec 02 2007
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MAPLE
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ZL := [S, {S=Prod(B, B, B, B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=8..40); # Zerinvary Lajos, Mar 13 2007
A000581:=-1/(z-1)**9; # Simon Plouffe in his 1992 dissertation, with offset 0
seq(binomial(n, 8), n=8..40); # Zerinvary Lajos, Jun 23 2008
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MATHEMATICA
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Table[Binomial[n, 8], {n, 8, 50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
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PROG
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(MAGMA) [Binomial(n, 8): n in [8..100]]; // Vincenzo Librandi, Apr 08 2011
(PARI) a(n)=binomial(n, 8) \\ Charles R Greathouse IV, Feb 14 2012
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CROSSREFS
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Cf. A000217, A000292, A000332, A000389, A000579, A000580, A053130, A053137, A254142.
Sequence in context: A289354 A008491 A023034 * A306939 A229889 A243743
Adjacent sequences: A000578 A000579 A000580 * A000582 A000583 A000584
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000
Some formulas referring to other offsets rewritten by R. J. Mathar, Jul 07 2009
3 more terms from William Boyles, Aug 06 2015
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STATUS
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approved
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